Editing Relevance logic (section)

===Algebraic models===
Some relevance logics can be given algebraic models, such as the logic R. The algebraic structures for R are de Morgan monoids, which are sextuples <math>(D,\land,\lor,\lnot,\circ,e)</math> where
* <math>(D,\land,\lor,\lnot)</math> is a distributive [[Lattice (order)|lattice]] with a unary operation, <math>\lnot</math> obeying the laws <math>\lnot\lnot x=x</math> and if <math>x\leq y</math> then <math>\lnot y\leq \lnot x</math>;
* <math>e\in D</math>, the binary operation <math>\circ</math> is [[Commutative property|commutative]] (<math>x\circ y=y\circ x</math>) and [[Associative property|associative]] (<math>(x\circ y)\circ z=x\circ (y\circ z)</math>), and <math>e\circ x=x</math>, i.e. <math>(D,\circ,e)</math> is an [[Monoid#Commutative monoid|Abelian monoid]] with [[Identity element|identity]] <math>e</math>;
* the monoid is lattice-ordered and satisfies <math>x\circ(y\lor z)=(x\circ y)\lor(x\circ z)</math>;
* <math>x\leq x\circ x</math>; and 
* if <math>x\circ y\leq z</math>, then <math>x\circ\lnot z\leq \lnot y</math>.
The operation <math>x\to y</math> interpreting the conditional of R is defined as <math>\lnot(x\circ\lnot y)</math>. 
A de Morgan monoid is a [[residuated lattice]], obeying the following residuation condition. 
: <math>x \circ y\leq z \iff x\leq y\to z</math>

An interpretation <math>v</math> is a [[homomorphism]] from the propositional language to a de Morgan monoid <math>M</math> such that 
* <math>v(p)\in D</math> for all atomic propositions, 
* <math>v(\lnot A)=\lnot v(A)</math>
* <math>v(A\lor B)=v(A)\lor v(B)</math>
* <math>v(A\land B)=v(A)\land v(B)</math>
* <math>v(A\to B)=v(A)\to v(B)</math>

Given a de Morgan monoid <math>M</math> and an interpretation <math>v</math>, one can say that formula <math>A</math> holds on <math>v</math> just in case <math>e\leq v(A)</math>. A formula <math>A</math> is valid just in case it holds on all interpretations on all de Morgan monoids. The logic R is sound and complete for de Morgan monoids.